Exotic antidecuplet baryons are predicted to be not only surprisingly light
but also very narrow. First we explain how small decay width arises in the
quark soliton model. Next, we study possible mixing of exotic antidecuplet with
Roper octet and discuss its phenomenological consequences.Comment: 7 pages, 2 figures, presented at the workshop "Excited QCD 2010",
  Jan.31 - Feb.6, 2010, Stara Lesna, Slovakia. v3 replaces unsuccesful upload
  of v2. Referencing has been changed and some statements have been made more
  precise

Praszalowicz, Michal

Acta Physica Polonica B Proceedings Supplement

English

arXiv

Exotic antidecuplet baryons are predicted to be not only surprisingly light but also very narrow. First, we explain how small decay width arises in the quark soliton model. Next, we study possible mixing of exotic antidecuplet with Roper octet and discuss its phenomenological consequences

Praszałowicz, Michał

Jagiellonian Univeristy Repository

Vol. 3 (2010) Acta Physica Polonica B Proceedings Supplement No 4IMPORTANCE OF MIXING FOR EXOTIC BARYONS∗Michal PraszalowiczM. Smoluchowski Institute of Physics, Jagellonian UniversityReymonta 4, 30-059 Kraków, Poland(Received August 9, 2010)Exotic antidecuplet baryons are predicted to be not only surprisinglylight but also very narrow. First, we explain how small decay width arisesin the quark soliton model. Next, we study possible mixing of exotic antide-cuplet with Roper octet and discuss its phenomenological consequences.PACS numbers: 11.30.Rd, 12.39.Dc, 13.30.Eg, 14.20.–c1. IntroductionDespite recent skepticism concerning early announcements of the dis-covery of exotic strange baryon Θ+(1540) two collaborations DIANA andLEPS confirmed their original results [1–3]. We refer the reader to recentexperimental reviews [4–6]. In this article following [7] we assume that Θ+exists with a mass equal 1540 MeV and total width Γ < 1 MeV. An imme-diate consequences is the existence of the whole exotic SU(3) multiplet: 10.Apart from truly exotic states antidecuplet contains cryptoexotic nucleon-(N10) and Σ-like states (Σ10). The interpretation of these states is not wellunderstood: one may try to associate them with some known resonances, orone may postulate the existence of new resonances with nucleon or Σ quan-tum numbers. This is the approach which we adopt here [7]. FollowingRefs. [8–12] we assume that there exists new, narrow nucleon resonanceN(1685) which we will interpret as a member of 10. If so, N10 decays haveto satisfy the following constraints [7] originally discussed in [8]ΓN10→πN <0.5 MeV , BrN10→ηN >0.2 , 5 MeV < ΓtotN10<25 MeV . (1)Unfortunately, the small partial width of N(1685) to πN contradicts SU(3)symmetry relations between the decay constants of N10.∗ Presented at the Workshop “Excited QCD 2010”, Tatranská Lomnica/Stará Lesná,Tatra National Park, Slovakia, January 31–February 6, 2010.(917)918 M. PraszalowiczIn this short note we would like to emphasize the importance of mix-ing both for decays and mass spectra of the putative exotic antidecupletbaryons. The exotic states have been already anticipated by the foundersof the quark model although they did not elaborate on them. Later the ar-guments have been raised that they should be heavy and wide. In contrast,chiral soliton models predicted that the pentaquark masses were genericallysmall (i.e. in the range of 1.5–1.6 GeV) [13–15]. It was much more difficultto accommodate the small decay width of Θ+ [16]. In Sec. 2 we explainhow small decay width arises naturally in the Chiral Soliton Quark Model(χQSM). Next, in Sec. 3 we argue that the decay coupling gΘNK is furtherreduced due to Gell-Mann–Okubo (GMO) mixing caused by the nonzeroms.In Sec. 4 we show how additional mixing of 10 with Roper octet can changedecay patterns of N10. We estimate allowed range of mixing angles andpresent predictions for remaining members of 10: Σ10 and Ξ10. Conclusionsare presented in Sec. 5.2. Decay widths in chiral soliton quark modelIn in Ref. [15] the following nonrelativistic formula for the decay widthhas been usedΓB1→B2ϕ =g2B1B2ϕ2π(M1 +M2)2p3ϕ . (2)It follows from the Goldberger–Treimann relation between axial and strongdecay constants. Here M1 is the mass of the decaying baryon, M2 the massof the decay product and pϕ is the outgoing meson momentum. GenericallyΓΘ+→NK given by (2) would be still in the range of a few hundreds ofMeV [16] if not for the terms non-leading in 1/Nc expansion. Indeed, evenwithout mixing the decay constant gB1B2ϕ, which stands for the matrixelement of the tensor decay operator O(8)ϕ between the physical baryon wavefunctions∣∣Bphys〉gB1B2ϕ =〈Bphys2∣∣∣O(8)ϕ ∣∣∣Bphys1 〉 (3)is the sum of three different contributions that are formally of differentorder in 1/Nc. However, they are multiplied by the SU(3) Clebsch–Gordancoefficients that also depend on Nc [17]. For example,g2ΘNK =9(Nc + 1)(Nc + 3)(Nc + 7)G210with G10 = G0 −Nc + 14G1 − 12 G2 , (4)Importance of Mixing for Exotic Baryons 919where G0 ∼ N3/2c , G1,2 ∼ N1/2c . Similarlyg2∆Nπ =3(Nc − 1)(Nc + 5)2(Nc + 1)(Nc + 7)G210 , with G10 = G0 +12 G2 . (5)Chiral soliton models provide us with specific predictions for constantsG0,1,2 [18]. Had we neglected G1 and G2 in Eqs. (4,5) (which would beinconsistent for gΘNK because of Nc enhancement of G1) we would haveobtained for Nc = 3g∆Nπ = gΘNK ∼ 17.6estimating G0 from the experimental value of ∆ decay width, and conse-quentlyΓΘNK ∼ 150 MeV . (6)We see that small decay width of Θ+ results from the cancellation in (4).Indeed, the authors of Ref. [15] have shown that in the nonrelativistic limitof χQSM one obtains that G0 =−(Nc + 2)G, G1 =−4G, G2 =−2G, withG ∼ N1/2c and consequently ΓΘNK = 0! In the same limit χQSM pre-dicts that gA = 3/5 and µp/µn = −3/2. It follows that antidecuplet decayconstants are small.3. Gell-Mann–Okubo mixingTreating ms corrections as perturbation introduces mixing [7]∣∣∣Bphys8 〉 = ∣∣81/2, B〉+cB10 ∣∣101/2, B〉+cB27 ∣∣271/2, B〉 ,∣∣∣Bphys10〉=∣∣101/2, B〉+dB8 ∣∣81/2, B〉+dB27 ∣∣271/2, B〉+dB35 ∣∣351/2, B〉 , (7)where subscripts refer to spin. For some specific states mixing constantscBR, dBR ∼ ms may be equal zero due to the isospin. For example Θ+ mixesonly with 35, but this component of the wave function does not contributeto the decays to octet. Therefore only mixing of the final nucleon with 10and 27 modifies the decay constantg2ΘNK =35[G10 +54c10H10 −74c27H′27]2. (8)Since G10 is small the admixtures proportional to the reduced matrix ele-mentsH10 ∼〈101/2, B′∣∣ Ô(8)ϕ ∣∣101/2, B〉 ,H ′27 ∼〈271/2, B′∣∣ Ô(8)ϕ ∣∣101/2, B〉 (9)920 M. Praszalowiczare important even if mixing parameters c10 and c27 are not large (for def-initions see Ref. [19]). Neither H10 nor H′27 vanish in the nonrelativisticlimit. Therefore in this limit Θ+ decay occurs entirely due to the mixing.For realistic model parameters (when G10 > 0) there is a cancellation (notethat H10 < 0 and H′27 > 0) between different terms in (8) and gΘNK isfurther suppressed. Mixing affects decay patterns of pentaquarks violatingSU(3) relations between the decay constants gB1B2ϕ [19].On somewhat more phenomenological ground let us consider only 8↔ 10mixing [7] defininggθNK = cosα g10 + sinαh10 , (10)which can be directly extracted from (2) if ΓΘNK is known. Throughoutthis paper we assume that ΓΘNK ' 1 MeV, hence gθNK ' 1.4. It is thenpossible to express decay constants of other members of antidecuplet interms of measurable physical parameters such as gπNN ' 13.2, ε = F/D '0.56, mixing angle α and one a priori unknown parameter h10 which can beestimated from χQSM calculations. Here we take h10 = −7 [7]. Then wehave for example,gN10Nπ =12 cosα gθNK − tanα√3gπNN ,gN10Nη =12 cosα gθNK −12 sin 2αh10 + tanα3ε− 11 + εgπNN√3. (11)If we want to interpret N(1685) as N10 we need to satisfy bounds (1).This is quite difficult within one angle scenario. It is possible to nullifygN10Nπ by a suitable choice of mixing angle α ∼ 0.03, but then the meanoctet mass (i.e. the nucleon mass before mixing) which is experimentally1151MeV comes out wrong [7]M8 = MphysN cos2 α+M2N10 sin2 α 'MphysN . (12)The mixing angle that satisfies (1) is an order of magnitude too small toaccount for baryon masses. For realistic mixing angles gN10Nπ is dominatedby gπNN and ΓN10Nπ > ΓN10Nη in contradiction with experimental data forN(1685).4. Mixing with RoperIn order to satisfy conditions (1) with more realistic mixing angles wehave considered in Ref. [7] scenario in which exotic antidecuplet can mixwith Roper resonance octet (mixing angle φ). For Roper octet GMO massformulae work with much worse accuracy than for the ground state octet[20, 21], so there is a need for additional mixing. Since εRoper ∼ 1/3 [21]Importance of Mixing for Exotic Baryons 921Roper decay to Nη is negligible. However Roper admixture contributes toNπ and other decay modesgN10Nπ =12 cosφ cosα gθNK − cosφ tanα√3gπNN − tanφ gRNπ ,gN10Nη =12 cosφ cosα gθNK −12 cosφ sin 2αh10+ cosφ tanα3ε− 11 + εgπNN√3. (13)Since gRNπ ∼ 12 is comparable with gπNN one may suppress gN10Nπ withoutchanging much gN10Nη. In Ref. [7] we have found that conditions (1) aresatisfied in the vicinity of the lineφ(α) = 0.0508− 2.207α , 0.079 < α < 0.159 . (14)We see that mixing angles are reasonable. The decay widths and branchingratio to Nη along (14) are plotted in Fig. 1.GHNΠL @MeVDBrHNΗLGtot  25 @MeVD0.00 0.05 0.10 0.15 0.20Α0.20.40.60.81.0Br or GFig. 1. Total, solid black line, decay width (divided by 25) and partial decay widthof N10 to πN , long dashed (red) line in MeV together with branching ratio, shortdashed (blue) line of N10 → ηN as functions of mixing angle α along the line (14).Thin vertical lines correspond to the limits on the mixing angle α. The plot ismade for h10 = −7.In Fig. 2 we plot gN10Nπ and gN10Nη together with their different com-ponents along the line (14). We see that indeed gN10Nπ is small due to thecancellation between gπNN and gRNπ, while gN10Nη rises moderately whenmixing increases.Having established the range of mixing angles we can predict masses ofthe remaining antidecuplet members [7]1795 MeV < MΣ10 < 1830 MeV , (15)1900 MeV < MΞ10 < 1970 MeV . (16)922 M. PraszalowiczgRNΠgQNKgΠNN0.05 0.10 0.15 0.20Α-3-2-1123gNNΠgQNKh10gΠNN0.00 0.05 0.10 0.15 0.20Α0.51.01.52.02.53.0gNNΗFig. 2. Decay constants of N10 (solid lines) with different components defined inEq. (13) shown by dashed lines. The plot is made for h10 = −7.Note that bound (16) contradicts the result of NA49MΞ10 ∼ 1860 MeV [22].From our analysis it follows that total decay width of Ξ10 to KΣ and πΞis of the order of 10MeV. Total width of Σ10 does not exceed 30 MeV butis also constrained from below to be larger that 10 MeV. Most prominentdecay channels are KN and πΛ with branching ratios approximately 60%and 20%, respectively. Due to the mixing SU(3) forbidden decays to decupletare possible, but small, at the level of 5 to 9%.5. Summary and conclusionsMixing induced by ms was first studied within the χQSM already inRef. [15] but only in the leading order in Nc. It was extended to nonlead-ing terms in Ref. [8] and [19]. Mixing appears also in other approaches topentaquarks. For example, in a diquark model [23] antidecuplet mixes withan accompanying cryptoexotic octet. Diagonalization of strangeness inducesideal (large) mixing between these two representations. The resulting phys-ical states of nucleon quantum numbers have been interpreted as Roper andN∗(1710). However, due to the ideal mixing these two states should havecomparable widths [24], while experimentally they differ substantially. Thediscussion of masses and decay widths of the N∗ states under assumptionthat they correspond to the Roper and N∗(1710) done in Ref. [25] still in-dicates that it is impossible to match the mass splittings with the observedbranching ratios for these two resonances even for arbitrary mixing. Whetherany different assignment of the diquark N∗ states would be compatible withthe decay patterns deserves a separate study.In this short note we have argued that due to the smallness of the re-duced matrix elements of 10→ 8, which is natural in chiral soliton models,mixing with other SU(3) representations has to be taken into account. Un-fortunately, for the time being we can only speculate which mixing scenariois phenomelogically impossible, allowed or desired. 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Importance of Mixing for Exotic Baryons

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